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C5×AGL(1,𝔽5)×C43⋊C3 #27166402

order12900
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productC5×AGL(1,𝔽5)×C43⋊C3
unambiguous semidirect product(AGL(1,𝔽5)×C43)⋊(C3×C5)
unambiguous semidirect product(C5×AGL(1,𝔽5)×C43)⋊C3

Splittings

Fitting, socle, HallC52×C43 ⋊ C3×C4
derivedC5×C43 ⋊ C3×C4×C5
HallC5×AGL(1,𝔽5)×C43 ⋊ C3
HallC52×C43⋊C3 ⋊ C4
SylowC43 ⋊ C3×C5×AGL(1,𝔽5)
SylowC52 ⋊ C4×C43⋊C3

Subgroups

subgroupG/H
center, hypercenterC5AGL(1,𝔽5)×C43⋊C3
commutator, hypocenterC5×C43C3×C4×C5
FrattiniC1G
Fitting, socleC52×C43C3×C4
prime radicalC5×D5×C43⋊C3C2
cosocleD5×C43C2×C3×C5
LCSG ▷ C5×C43
UCSC1 ◁ C5

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 53
transitive degreeμt(G) = 860

A minimal faithful permutation representation — 2 generators on 53 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,33)(18,26,31)(21,29,28)(23,32,25)(27,36,40)(30,37,39)(34,41,35)(38,42,43)(44,45,46,47,48)(50,51,52,53);(1,2,4,8,15,20,5,9,16,25,35,32,12,19,28,24,34,41,22,31,39,36,14,23,33,40,43,42,26,7,13,21,30,38,37,17,3,6,11,18,27,29,10)(49,50,51,53,52)

Conjugacy classes & character table

425 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 71);